PART rail moment of inertia

					                                       Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 1


      Serway/Jewett, Physics for Scientists and Engineers, 8/e
                    PSE 8e - Chapter 15 Oscillatory Motion
                              Questions and Problems



Objective Questions
      denotes answer available in Student Solutions Manual / Study
Guide


1. The position of an object moving with simple harmonic
motion is given by x  4cos  6 t  , where x is in meters and t is

in seconds. What is the period of the oscillating system? (a) 4 s
(b)   1
      6
          s   (c)   1
                    3
                        s   (d) 6 s    (e) impossible to determine from the

information given


2. Which of the following statements is not true regarding a
mass–spring system that moves with simple harmonic motion
in the absence of friction? (a) The total energy of the system
remains constant. (b) The energy of the system is continually
transformed between kinetic and potential energy. (c) The total
energy of the system is proportional to the square of the
amplitude. (d) The potential energy stored in the system is
greatest when the mass passes through the equilibrium
position. (e) The velocity of the oscillating mass has its
maximum value when the mass passes through the equilibrium
                                Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 2


position.


3. A block–spring system vibrating on a frictionless, horizontal
surface with an amplitude of 6.0 cm has an energy of 12 J. If the
block is replaced by one whose mass is twice the mass of the
original block and the amplitude of the motion is again 6.0 cm,
what is the energy of the system? (a) 12 J            (b) 24 J (c) 6 J   (d)
48 J   (e) none of those answers


4. If an object of mass m attached to a light spring is replaced by
one of mass 9m, the frequency of the vibrating system changes
by what factor? (a)   1
                      9   (b)   1
                                3
                                    (c) 3.0 (d) 9.0 (e) 6.0


5. An object of mass 0.40 kg, hanging from a spring with a
spring constant of 8.0 N/m, is set into an up-and-down simple
harmonic motion. What is the magnitude of the acceleration of
the object when it is at its maximum displacement of 0.10 m?
(a) zero (b) 0.45 m/s2 (c) 1.0 m/s2 (d) 2.0 m/s2 (e) 2.4 m/s2


6. A runaway railroad car, with mass 3.0  105 kg, coasts across
a level track at 2.0 m/s when it collides elastically with a
spring-loaded bumper at the end of the track. If the spring
constant of the bumper is 2.0  106 N/m, what is the maximum
compression of the spring during the collision? (a) 0.77 m (b)
0.58 m (c) 0.34 m (d) 1.07 m (e) 1.24 m
                            Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 3




7. If a simple pendulum oscillates with small amplitude and its
length is doubled, what happens to the frequency of its motion?

(a) It doubles. (b) It becomes   2 times as large. (c) It becomes

half as large. (d) It becomes 1/ 2 times as large. (e) It remains
the same.


8. An object–spring system moving with simple harmonic
motion has an amplitude A. When the kinetic energy of the
object equals twice the potential energy stored in the spring,

what is the position x of the object? (a) A (b)    1
                                                   3
                                                       A (c) A   3 (d) 0

(e) none of those answers


9. A particle on a spring moves in simple harmonic motion
along the x axis between turning points at x1 = 100 cm and x2 =
140 cm. (i) At which of the following positions does the particle
have maximum speed? (a) 100 cm (b) 110 cm (c) 120 cm (d) at
none of those positions (ii) At which position does it have
maximum acceleration? Choose from the same possibilities as
in part (i). (iii) At which position is the greatest net force
exerted on the particle? Choose from the same possibilities as in
part (i).


10. A mass–spring system moves with simple harmonic motion
along the x axis between turning points at x1 = 20 cm and x2 =
                           Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 4


60 cm. For parts (i) through (iii), choose from the same five
possibilities. (i) At which position does the particle have the
greatest magnitude of momentum? (a) 20 cm (b) 30 cm (c) 40
cm (d) some other position (e) The greatest value occurs at
multiple points. (ii) At which position does the particle have
greatest kinetic energy? (iii) At which position does the
particle-spring system have the greatest total energy?


11. A block with mass m = 0.1 kg oscillates with amplitude A =
0.1 m at the end of a spring with force constant k = 10 N/m on a
frictionless, horizontal surface. Rank the periods of the
following situations from greatest to smallest. If any periods
are equal, show their equality in your ranking. (a) The system
is as described above. (b) The system is as described in
situation (a) except the amplitude is 0.2 m. (c) The situation is
as described in situation (a) except the mass is 0.2 kg. (d) The
situation is as described in situation (a) except the spring has
force constant 20 N/m. (e) A small resistive force makes the
motion underdamped.


12. For a simple harmonic oscillator, answer yes or no to the
following questions. (a) Can the quantities position and
velocity have the same sign? (b) Can velocity and acceleration
have the same sign? (c) Can position and acceleration have the
same sign?
                             Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 5




13. The top end of a spring is held fixed. A block is hung on the
bottom end as in Figure OQ15.13a, and the frequency f of the
oscillation of the system is measured. The block, a second
identical block, and the spring are carried up in a space shuttle
to Earth orbit. The two blocks are attached to the ends of the
spring. The spring is compressed without making adjacent coils
touch (Fig. OQ15.13b), and the system is released to oscillate
while floating within the shuttle cabin (Fig. OQ15.13c). What is
the frequency of oscillation for this system in terms of f? (a) f/2

(b) f / 2 (c) f (d)   2 f (e) 2f




14. You attach a block to the bottom end of a spring hanging
vertically. You slowly let the block move down and find that it
hangs at rest with the spring stretched by 15.0 cm. Next, you lift
the block back up to the initial position and release it from rest
with the spring unstretched. What maximum distance does it
move down? (a) 7.5 cm (b) 15.0 cm (c) 30.0 cm (d) 60.0 cm (e)
The distance cannot be determined without knowing the mass
                             Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 6


and spring constant.


15. A simple pendulum has a period of 2.5 s. (i) What is its
period if its length is made four times larger? (a) 1.25 s (b) 1.77 s
(c) 2.5 s (d) 3.54 s (e) 5 s (ii) What is its period if the length is
held constant at its initial value and the mass of the suspended
bob is made four times larger? Choose from the same
possibilities.


16. A simple pendulum is suspended from the ceiling of a
stationary elevator, and the period is determined. (i) When the
elevator accelerates upward, is the period (a) greater, (b)
smaller, or (c) unchanged? (ii) When the elevator has a
downward acceleration, is the period (a) greater, (b) smaller, or
(c) unchanged? (iii) When the elevator moves with constant
upward velocity, is the period of the pendulum (a) greater, (b)
smaller, or (c) unchanged?


17. You stand on the end of a diving board and bounce to set it
into oscillation. You find a maximum response in terms of the
amplitude of oscillation of the end of the board when you
bounce at frequency f. You now move to the middle of the
board and repeat the experiment. Is the resonance frequency
for forced oscillations at this point (a) higher, (b) lower, or (c)
the same as f?
                           Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 7




Conceptual Questions
   denotes answer available in Student Solutions Manual / Study
Guide


1. Is a bouncing ball an example of simple harmonic motion? Is
the daily movement of a student from home to school and back
simple harmonic motion? Why or why not?


2. The equations listed in Table 2.2 give position as a function
of time, velocity as a function of time, and velocity as a function
of position for an object moving in a straight line with constant
acceleration. The quantity vxi appears in every equation. (a) Do
any of these equations apply to an object moving in a straight
line with simple harmonic motion? (b) Using a similar format,
make a table of equations describing simple harmonic motion.
Include equations giving acceleration as a function of time and
acceleration as a function of position. State the equations in
such a form that they apply equally to a block–spring system,
to a pendulum, and to other vibrating systems. (c) What
quantity appears in every equation?


3. (a) If the coordinate of a particle varies as x = –A cos t, what
is the phase constant in Equation 15.6? (b) At what position is
the particle at t = 0?
                             Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 8




4. A simple pendulum can be modeled as exhibiting simple
harmonic motion when  is small. Is the motion periodic when
 is large?


5. Figure CQ15.5 shows graphs of the potential energy of four
different systems versus the position of a particle in each
system. Each particle is set into motion with a push at an
arbitrarily chosen location. Describe its subsequent motion in
each case (a), (b), (c), and (d).




6. A student thinks that any real vibration must be damped. Is
the student correct? If so, give convincing reasoning. If not, give
an example of a real vibration that keeps constant amplitude
forever if the system is isolated.


7. The mechanical energy of an undamped block–spring system
is constant as kinetic energy transforms to elastic potential
energy and vice versa. For comparison, explain what happens
                            Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 9


to the energy of a damped oscillator in terms of the mechanical,
potential, and kinetic energies.


8. Is it possible to have damped oscillations when a system is at
resonance? Explain.


9. Will damped oscillations occur for any values of b and k?
Explain.


10. If a pendulum clock keeps perfect time at the base of a
mountain, will it also keep perfect time when it is moved to the
top of the mountain? Explain.


11. You are looking at a small, leafy tree. You do not notice any
breeze, and most of the leaves on the tree are motionless. One
leaf, however, is fluttering back and forth wildly. After a while,
that leaf stops moving and you notice a different leaf moving
much more than all the others. Explain what could cause the
large motion of one particular leaf.


12. A pendulum bob is made from a sphere filled with water.
What would happen to the frequency of vibration of this
pendulum if there were a hole in the sphere that allowed the
water to leak out slowly?
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 10


13. Consider the simplified single-piston engine in Figure
CQ15.13. Assuming the wheel rotates with constant angular
speed, explain why the piston rod oscillates in simple harmonic
motion.




Problems

              The problems found in this chapter may be
assigned online in Enhanced WebAssign.
1. denotes straightforward problem; 2. denotes intermediate
problem; 3. denotes challenging problem
1. full solution available in the Student Solutions Manual/ Study
Guide
1. denotes problems most often assigned in Enhanced
WebAssign; these provide students with targeted feedback and
either a Master It tutorial or a Watch It solution video.
Q|C denotes   asking for quantitative and conceptual reasoning

    denotes symbolic reasoning problem
   denotes Master It tutorial available in Enhanced WebAssign
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 11


    denotes guided problem
shaded denotes "paired problems" that develop reasoning
with symbols and numeric values


Note: Ignore the mass of every spring, except in Problems 64
and 75.


Section 15.1 Motion of an Object Attached to a Spring


Problems 16, 17, 18, 22, and 65 in Chapter 7 can also be
assigned with this section.


1. A 0.60-kg block attached to a spring with force constant 130
N/m is free to move on a frictionless, horizontal surface as in
Active Figure 15.1. The block is released from rest when the
spring is stretched 0.13 m. At the instant the block is released,
find (a) the force on the block and (b) its acceleration.


2. When a 4.25-kg object is placed on top of a vertical spring,
the spring compresses a distance of 2.62 cm. What is the force
constant of the spring?


Section 15.2 Analysis Model: Particle in Simple Harmonic Motion


3. A vertical spring stretches 3.9 cm when a 10-g object is hung
                            Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 12


from it. The object is replaced with a block of mass 25 g that
oscillates up and down in simple harmonic motion. Calculate
the period of motion.


4. In an engine, a piston oscillates with simple harmonic motion
so that its position varies according to the expression
                                           
                         x = 5.00 cos  2t  
                                           6
where x is in centimeters and t is in seconds. At t = 0, find (a)
the position of the particle, (b) its velocity, and (c) its
acceleration. Find (d) the period and (e) the amplitude of the
motion.


5. The position of a particle is given by the expression x = 4.00
cos (3.00t + ), where x is in meters and t is in seconds.
Determine (a) the frequency and (b) period of the motion, (c)
the amplitude of the motion, (d) the phase constant, and (e) the
position of the particle at t = 0.250 s.


6. A piston in a gasoline engine is in simple harmonic motion.
The engine is running at the rate of 3 600 rev/min. Taking the
extremes of its position relative to its center point as ±5.00 cm,
find the magnitudes of the (a) maximum velocity and (b)
maximum acceleration of the piston.
                           Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 13


7. A 1.00-kg object is attached to a horizontal spring. The spring
is initially stretched by 0.100 m, and the object is released from
rest there. It proceeds to move without friction. The next time
the speed of the object is zero is 0.500 s later. What is the
maximum speed of the object?


8. A simple harmonic oscillator takes 12.0 s to undergo five
complete vibrations. Find (a) the period of its motion, (b) the
frequency in hertz, and (c) the angular frequency in radians per
second.


9. A 7.00-kg object is hung from the bottom end of a vertical
spring fastened to an overhead beam. The object is set into
vertical oscillations having a period of 2.60 s. Find the force
constant of the spring.


10. Q|C (a) A hanging spring stretches by 35.0 cm when an
object of mass 450 g is hung on it at rest. In this situation, we
define its position as x = 0. The object is pulled down an
additional 18.0 cm and released from rest to oscillate without
friction. What is its position x at a moment 84.4 s later? (b) Find
the distance traveled by the vibrating object in part (a). (c)
What If? Another hanging spring stretches by 35.5 cm when an
object of mass 440 g is hung on it at rest. We define this new
position as x = 0. This object is also pulled down an additional
                           Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 14


18.0 cm and released from rest to oscillate without friction. Find
its position 84.4 s later. (d) Find the distance traveled by the
object in part (c). (e) Why are the answers to parts (a) and (c) so
different when the initial data in parts (a) and (c) are so similar
and the answers to parts (b) and (d) are relatively close? Does
this circumstance reveal a fundamental difficulty in calculating
the future?


11. Review. A particle moves along the x axis. It is initially at
the position 0.270 m, moving with velocity 0.140 m/s and
acceleration 0.320 m/s2. Suppose it moves as a particle under
constant acceleration for 4.50 s. Find (a) its position and (b) its
velocity at the end of this time interval. Next, assume it moves
as a particle in simple harmonic motion for 4.50 s and x = 0 is
its equilibrium position. Find (c) its position and (d) its velocity
at the end of this time interval.


12. Q|C A ball dropped from a height of 4.00 m makes an elastic
collision with the ground. Assuming no mechanical energy is
lost due to air resistance, (a) show that the ensuing motion is
periodic and (b) determine the period of the motion. (c) Is the
motion simple harmonic? Explain.


13. A particle moving along the x axis in simple harmonic
motion starts from its equilibrium position, the origin, at t = 0
                           Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 15


and moves to the right. The amplitude of its motion is 2.00 cm,
and the frequency is 1.50 Hz. (a) Find an expression for the
position of the particle as a function of time. Determine (b) the
maximum speed of the particle and (c) the earliest time (t > 0)
at which the particle has this speed. Find (d) the maximum
positive acceleration of the particle and (e) the earliest time (t >
0) at which the particle has this acceleration. (f) Find the total
distance traveled by the particle between t = 0 and t = 1.00 s.


14. A 1.00-kg glider attached to a spring with a force constant of
25.0 N/m oscillates on a frictionless, horizontal air track. At t =
0, the glider is released from rest at x = –3.00 cm (that is, the
spring is compressed by 3.00 cm). Find (a) the period of the
glider’s motion, (b) the maximum values of its speed and
acceleration, and (c) the position, velocity, and acceleration as
functions of time.


15. A 0.500-kg object attached to a spring with a force constant
of 8.00 N/m vibrates in simple harmonic motion with an
amplitude of 10.0 cm. Calculate the maximum value of its (a)
speed and (b) acceleration, (c) the speed and (d) the
acceleration when the object is 6.00 cm from the equilibrium
position, and (e) the time interval required for the object to
move from x = 0 to x = 8.00 cm.
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 16


16. Q|C You attach an object to the bottom end of a hanging
vertical spring. It hangs at rest after extending the spring 18.3
cm. You then set the object vibrating. (a) Do you have enough
information to find its period? (b) Explain your answer and
state whatever you can about its period.


Section 15.3 Energy of the Simple Harmonic Oscillator


17.     To test the resiliency of its bumper during low-speed
collisions, a 1 000-kg automobile is driven into a brick wall. The
car’s bumper behaves like a spring with a force constant 5.00 
106 N/m and compresses 3.16 cm as the car is brought to rest.
What was the speed of the car before impact, assuming no
mechanical energy is transformed or transferred away during
impact with the wall?


18. A 200-g block is attached to a horizontal spring and
executes simple harmonic motion with a period of 0.250 s. The
total energy of the system is 2.00 J. Find (a) the force constant of
the spring and (b) the amplitude of the motion.


19. A 50.0-g object connected to a spring with a force constant
of 35.0 N/m oscillates with an amplitude of 4.00 cm on a
frictionless, horizontal surface. Find (a) the total energy of the
system and (b) the speed of the object when its position is 1.00
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 17


cm. Find (c) the kinetic energy and (d) the potential energy
when its position is 3.00 cm.


20. A 2.00-kg object is attached to a spring and placed on a
frictionless, horizontal surface. A horizontal force of 20.0 N is
required to hold the object at rest when it is pulled 0.200 m
from its equilibrium position (the origin of the x axis). The
object is now released from rest from this stretched position,
and it subsequently undergoes simple harmonic oscillations.
Find (a) the force constant of the spring, (b) the frequency of the
oscillations, and (c) the maximum speed of the object. (d)
Where does this maximum speed occur? (e) Find the maximum
acceleration of the object. (f) Where does the maximum
acceleration occur? (g) Find the total energy of the oscillating
system. Find (h) the speed and (i) the acceleration of the object
when its position is equal to one-third the maximum value.


21. Q|C     A simple harmonic oscillator of amplitude A has a
total energy E. Determine (a) the kinetic energy and (b) the
potential energy when the position is one-third the amplitude.
(c) For what values of the position does the kinetic energy
equal one-half the potential energy? (d) Are there any values of
the position where the kinetic energy is greater than the
maximum potential energy? Explain.
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 18


22.     Review. A 65.0-kg bungee jumper steps off a bridge
with a light bungee cord tied to her body and to the bridge. The
unstretched length of the cord is 11.0 m. The jumper reaches the
bottom of her motion 36.0 m below the bridge before bouncing
back. We wish to find the time interval between her leaving the
bridge and her arriving at the bottom of her motion. Her
overall motion can be separated into an 11.0-m free fall and a
25.0-m section of simple harmonic oscillation. (a) For the free-
fall part, what is the appropriate analysis model to describe her
motion? (b) For what time interval is she in free fall? (c) For the
simple harmonic oscillation part of the plunge, is the system of
the bungee jumper, the spring, and the Earth isolated or non-
isolated? (d) From your response in part (c) find the spring
constant of the bungee cord. (e) What is the location of the
equilibrium point where the spring force balances the
gravitational force exerted on the jumper? (f) What is the
angular frequency of the oscillation? (g) What time interval is
required for the cord to stretch by 25.0 m? (h) What is the total
time interval for the entire 36.0-m drop?


23. Q|C Review. A 0.250-kg block resting on a frictionless,
horizontal surface is attached to a spring whose force constant

is 83.8 N/m as in Figure P15.23. A horizontal force          causes the
spring to stretch a distance of 5.46 cm from its equilibrium
position. (a) Find the magnitude of      . (b) What is the total
                           Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 19


energy stored in the system when the spring is stretched? (c)
Find the magnitude of the acceleration of the block just after the
applied force is removed. (d) Find the speed of the block when
it first reaches the equilibrium position. (e) If the surface is not
frictionless but the block still reaches the equilibrium position,
would your answer to part (d) be larger or smaller? (f) What
other information would you need to know to find the actual
answer to part (d) in this case? (g) What is the largest value of
the coefficient of friction that would allow the block to reach
the equilibrium position?




24. A 326-g object is attached to a spring and executes simple
harmonic motion with a period of 0.250 s. If the total energy of
the system is 5.83 J, find (a) the maximum speed of the object,
(b) the force constant of the spring, and (c) the amplitude of the
motion.


Section 15.4 Comparing Simple Harmonic Motion with
Uniform Circular Motion


25. Q|C While driving behind a car traveling at 3.00 m/s, you
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 20


notice that one of the car’s tires has a small hemispherical
bump on its rim as shown in Figure P15.25. (a) Explain why the
bump, from your viewpoint behind the car, executes simple
harmonic motion. (b) If the radii of the car’s tires are 0.300 m,
what is the bump’s period of oscillation?




Section 15.5 The Pendulum


Problem 62 in Chapter 1 can also be assigned with this section.


26. A ―seconds pendulum‖ is one that moves through its
equilibrium position once each second. (The period of the
pendulum is precisely 2 s.) The length of a seconds pendulum
is 0.992 7 m at Tokyo, Japan, and 0.994 2 m at Cambridge,
England. What is the ratio of the free-fall accelerations at these
two locations?


27. A simple pendulum makes 120 complete oscillations in 3.00
min at a location where g = 9.80 m/s2. Find (a) the period of the
pendulum and (b) its length.
                         Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 21




28.    A particle of mass m slides without friction inside a
hemispherical bowl of radius R. Show that if the particle starts
from rest with a small displacement from equilibrium, it moves
in simple harmonic motion with an angular frequency equal to

that of a simple pendulum of length R. That is,   g / R .


29. A physical pendulum in the form of a planar object moves
in simple harmonic motion with a frequency of 0.450 Hz. The
pendulum has a mass of 2.20 kg, and the pivot is located 0.350
m from the center of mass. Determine the moment of inertia of
the pendulum about the pivot point.


30.   A physical pendulum in the form of a planar object
moves in simple harmonic motion with a frequency f. The
pendulum has a mass m, and the pivot is located a distance d
from the center of mass. Determine the moment of inertia of the
pendulum about the pivot point.


31. Q|C A simple pendulum has a mass of 0.250 kg and a length
of 1.00 m. It is displaced through an angle of 15.0 and then
released. Using the analysis model of a particle in simple
harmonic motion, what are (a) the maximum speed of the bob,
(b) its maximum angular acceleration, and (c) the maximum
restoring force on the bob? (d) What If? Solve parts (a) through
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 22


(c) again by using analysis models introduced in earlier
chapters. (e) Compare the answers.


32.     Consider the physical pendulum of Figure 15.17. (a)
Represent its moment of inertia about an axis passing through
its center of mass and parallel to the axis passing through its
pivot point as ICM. Show that its period is

                               I CM  md 2
                        T  2
                                   mgd

where d is the distance between the pivot point and the center
of mass. (b) Show that the period has a minimum value when d
satisfies md2 = ICM.




33. Review. A simple pendulum is 5.00 m long. What is the
period of small oscillations for this pendulum if it is located in
an elevator (a) accelerating upward at 5.00 m/s2? (b)
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 23


Accelerating downward at 5.00 m/s2? (c) What is the period of
this pendulum if it is placed in a truck that is accelerating
horizontally at 5.00 m/s2?


34. A very light rigid rod of length 0.500 m extends straight out
from one end of a meterstick. The combination is suspended
from a pivot at the upper end of the rod as shown in Figure
P15.34. The combination is then pulled out by a small angle
and released. (a) Determine the period of oscillation of the
system. (b) By what percentage does the period differ from the
period of a simple pendulum 1.00 m long?




35. A watch balance wheel (Fig. P15.35) has a period of
oscillation of 0.250 s. The wheel is constructed so that its mass
of 20.0 g is concentrated around a rim of radius 0.500 cm. What
are (a) the wheel’s moment of inertia and (b) the torsion
constant of the attached spring?
                           Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 24




36. A small object is attached to the end of a string to form a
simple pendulum. The period of its harmonic motion is
measured for small angular displacements and three lengths.
For lengths of 1.000 m, 0.750 m, and 0.500 m, total time
intervals for 50 oscillations of 99.8 s, 86.6 s, and 71.1 s are
measured with a stopwatch. (a) Determine the period of motion
for each length. (b) Determine the mean value of g obtained
from these three independent measurements and compare it
with the accepted value. (c) Plot T2 versus L and obtain a value
for g from the slope of your best-fit straight-line graph. (d)
Compare the value found in part (c) with that obtained in part
(b).


Section 15.6 Damped Oscillations


37. A pendulum with a length of 1.00 m is released from an
initial angle of 15.0. After 1 000 s, its amplitude has been
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 25


reduced by friction to 5.50. What is the value of b/2m?


38.      Show that the time rate of change of mechanical energy
for a damped, undriven oscillator is given by dE/dt = –bv2 and
hence is always negative. To do so, differentiate the expression
for the mechanical energy of an oscillator, E  1 mv2  1 kx2 , and
                                                2       2

use Equation 15.31.


39. A 10.6-kg object oscillates at the end of a vertical spring that
has a spring constant of 2.05  104 N/m. The effect of air
resistance is represented by the damping coefficient b = 3.00 N ·
s/m. (a) Calculate the frequency of the damped oscillation. (b)
By what percentage does the amplitude of the oscillation
decrease in each cycle? (c) Find the time interval that elapses
while the energy of the system drops to 5.00% of its initial
value.


40.      Show that Equation 15.32 is a solution of Equation 15.31
provided that b2 < 4mk.


Section 15.7 Forced Oscillations


41. As you enter a fine restaurant, you realize that you have
accidentally brought a small electronic timer from home
instead of your cell phone. In frustration, you drop the timer
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 26


into a side pocket of your suit coat, not realizing that the timer
is operating. The arm of your chair presses the light cloth of
your coat against your body at one spot. Fabric with a length L
hangs freely below that spot, with the timer at the bottom. At
one point during your dinner, the timer goes off and a buzzer
and a vibrator turn on and off with a frequency of 1.50 Hz. It
makes the hanging part of your coat swing back and forth with
remarkably large amplitude, drawing everyone’s attention.
Find the value of L.


42. A baby bounces up and down in her crib. Her mass is 12.5
kg, and the crib mattress can be modeled as a light spring with
force constant 700 N/m. (a) The baby soon learns to bounce
with maximum amplitude and minimum effort by bending her
knees at what frequency? (b) If she were to use the mattress as
a trampoline—losing contact with it for part of each cycle—
what minimum amplitude of oscillation does she require?


43. A 2.00-kg object attached to a spring moves without friction
(b = 0) and is driven by an external force given by the
expression F = 3.00 sin (2t), where F is in newtons and t is in
seconds. The force constant of the spring is 20.0 N/m. Find (a)
the resonance angular frequency of the system, (b) the angular
frequency of the driven system, and (c) the amplitude of the
motion.
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 27




44. A block weighing 40.0 N is suspended from a spring that
has a force constant of 200 N/m. The system is undamped (b =
0) and is subjected to a harmonic driving force of frequency
10.0 Hz, resulting in a forced-motion amplitude of 2.00 cm.
Determine the maximum value of the driving force.


45.    Damping is negligible for a 0.150-kg object hanging
from a light, 6.30-N/m spring. A sinusoidal force with an
amplitude of 1.70 N drives the system. At what frequency will
the force make the object vibrate with an amplitude of 0.440 m?


46.    Considering an undamped, forced oscillator (b = 0),
show that Equation 15.35 is a solution of Equation 15.34, with
an amplitude given by Equation 15.36.


Additional Problems


47. The mass of the deuterium molecule (D2) is twice that of the
hydrogen molecule (H2). If the vibrational frequency of H2 is
1.30  1014 Hz, what is the vibrational frequency of D2? Assume
the ―spring constant‖ of attracting forces is the same for the
two molecules.
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 28


48. Q|C Review. This problem extends the reasoning of
Problem 59 in Chapter 9. Two gliders are set in motion on an
air track. Glider 1 has mass m1 = 0.240 kg and moves to the
right with speed 0.740 m/s. It will have a rear-end collision
with glider 2, of mass m2 = 0.360 kg, which initially moves to
the right with speed 0.120 m/s. A light spring of force constant
45.0 N/m is attached to the back end of glider 2 as shown in
Figure P9.59. When glider 1 touches the spring, superglue
instantly and permanently makes it stick to its end of the
spring. (a) Find the common speed the two gliders have when
the spring is at maximum compression. (b) Find the maximum
spring compression distance. The motion after the gliders
become attached consists of a combination of (1) the constant-
velocity motion of the center of mass of the two-glider system
found in part (a) and (2) simple harmonic motion of the gliders
relative to the center of mass. (c) Find the energy of the center-
of-mass motion. (d) Find the energy of the oscillation.




49. Q|C An object of mass m moves in simple harmonic motion
with amplitude 12.0 cm on a light spring. Its maximum
                           Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 29


acceleration is 108 cm/s2. Regard m as a variable. (a) Find the
period T of the object. (b) Find its frequency f. (c) Find the
maximum speed vmax of the object. (d) Find the total energy E of
the object–spring system. (e) Find the force constant k of the
spring. (f) Describe the pattern of dependence of each of the
quantities T, f, vmax, E, and k on m.


50. Q|C Review. A rock rests on a concrete sidewalk. An
earthquake strikes, making the ground move vertically in
simple harmonic motion with a constant frequency of 2.40 Hz
and with gradually increasing amplitude. (a) With what
amplitude does the ground vibrate when the rock begins to
lose contact with the sidewalk? Another rock is sitting on the
concrete bottom of a swimming pool full of water. The
earthquake produces only vertical motion, so the water does
not slosh from side to side. (b) Present a convincing argument
that when the ground vibrates with the amplitude found in
part (a), the submerged rock also barely loses contact with the
floor of the swimming pool.


51.    A small ball of mass M is attached to the end of a
uniform rod of equal mass M and length L that is pivoted at the
top (Fig. P15.51). Determine the tensions in the rod (a) at the
pivot and (b) at the point P when the system is stationary. (c)
Calculate the period of oscillation for small displacements from
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 30


equilibrium and (d) determine this period for L = 2.00 m.




52. An object attached to a spring vibrates with simple
harmonic motion as described by Figure P15.52. For this
motion, find (a) the amplitude, (b) the period, (c) the angular
frequency, (d) the maximum speed, (e) the maximum
acceleration, and (f) an equation for its position x as a function
of time.




53. Review. A large block P attached to a light spring executes
horizontal, simple harmonic motion as it slides across a
frictionless surface with a frequency f = 1.50 Hz. Block B rests
on it as shown in Figure P15.53, and the coefficient of static
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 31


friction between the two is s = 0.600. What maximum
amplitude of oscillation can the system have if block B is not to
slip?




54.     Review. A large block P attached to a light spring
executes horizontal, simple harmonic motion as it slides across
a frictionless surface with a frequency f. Block B rests on it as
shown in Figure P15.53, and the coefficient of static friction
between the two is s. What maximum amplitude of oscillation
can the system have if block B is not to slip?


55.     A pendulum of length L and mass M has a spring of
force constant k connected to it at a distance h below its point of
suspension (Fig. P15.55). Find the frequency of vibration of the
system for small values of the amplitude (small ). Assume the
vertical suspension rod of length L is rigid, but ignore its mass.
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 32




56. A particle with a mass of 0.500 kg is attached to a horizontal
spring with a force constant of 50.0 N/m. At the moment t = 0,
the particle has its maximum speed of 20.0 m/s and is moving
to the left. (a) Determine the particle’s equation of motion,
specifying its position as a function of time. (b) Where in the
motion is the potential energy three times the kinetic energy?
(c) Find the minimum time interval required for the particle to
move from x = 0 to x = 1.00 m. (d) Find the length of a simple
pendulum with the same period.


57. A horizontal plank of mass 5.00 kg and length 2.00 m is
pivoted at one end. The plank’s other end is supported by a
spring of force constant 100 N/m (Fig P15.57). The plank is
displaced by a small angle  from its horizontal equilibrium
position and released. Find the angular frequency with which
the plank moves with simple harmonic motion.
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 33




58.   A horizontal plank of mass m and length L is pivoted at
one end. The plank’s other end is supported by a spring of
force constant k (Fig P15.57). The plank is displaced by a small
angle  from its horizontal equilibrium position and released.
Find the angular frequency with which the plank moves with
simple harmonic motion.


59. Q|C Review. A particle of mass 4.00 kg is attached to a
spring with a force constant of 100 N/m. It is oscillating on a
frictionless, horizontal surface with an amplitude of 2.00 m. A
6.00-kg object is dropped vertically on top of the 4.00-kg object
as it passes through its equilibrium point. The two objects stick
together. (a) What is the new amplitude of the vibrating system
after the collision? (b) By what factor has the period of the
system changed? (c) By how much does the energy of the
system change as a result of the collision? (d) Account for the
change in energy.
                           Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 34


60. A simple pendulum with a length of 2.23 m and a mass of
6.74 kg is given an initial speed of 2.06 m/s at its equilibrium
position. Assume it undergoes simple harmonic motion.
Determine (a) its period, (b) its total energy, and (c) its
maximum angular displacement.


61. Review. One end of a light spring with force constant k =
100 N/m is attached to a vertical wall. A light string is tied to
the other end of the horizontal spring. As shown in Figure
P15.61, the string changes from horizontal to vertical as it
passes over a pulley of mass M in the shape of a solid disk of
radius R = 2.00 cm. The pulley is free to turn on a fixed, smooth
axle. The vertical section of the string supports an object of
mass m = 200 g. The string does not slip at its contact with the
pulley. The object is pulled downward a small distance and
released. (a) What is the angular frequency  of oscillation of
the object in terms of the mass M? (b) What is the highest
possible value of the angular frequency of oscillation of the
object? (c) What is the highest possible value of the angular
frequency of oscillation of the object if the pulley radius is
doubled to R = 4.00 cm?
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 35




62. Q|C People who ride motorcycles and bicycles learn to look
out for bumps in the road and especially for washboarding, a
condition in which many equally spaced ridges are worn into
the road. What is so bad about washboarding? A motorcycle
has several springs and shock absorbers in its suspension, but
you can model it as a single spring supporting a block. You can
estimate the force constant by thinking about how far the
spring compresses when a heavy rider sits on the seat. A
motorcyclist traveling at highway speed must be particularly
careful of washboard bumps that are a certain distance apart.
What is the order of magnitude of their separation distance?


63.    A ball of mass m is connected to two rubber bands of
length L, each under tension T as shown in Figure P15.63. The
ball is displaced by a small distance y perpendicular to the
length of the rubber bands. Assuming the tension does not
change, show that (a) the restoring force is – (2T/L) y and (b) the
system exhibits simple harmonic motion with an angular

frequency   2T / mL .
                           Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 36




64. When a block of mass M, connected to the end of a spring of
mass ms = 7.40 g and force constant k, is set into simple
harmonic motion, the period of its motion is

                                 M   ms / 3
                        T  2
                                      k
A two-part experiment is conducted with the use of blocks of
various masses suspended vertically from the spring as shown
in Figure P15.64. (a) Static extensions of 17.0, 29.3, 35.3, 41.3,
47.1, and 49.3 cm are measured for M values of 20.0, 40.0, 50.0,
60.0, 70.0, and 80.0 g, respectively. Construct a graph of Mg
versus x and perform a linear least-squares fit to the data. (b)
From the slope of your graph, determine a value for k for this
spring. (c) The system is now set into simple harmonic motion,
and periods are measured with a stopwatch. With M = 80.0 g,
the total time interval required for ten oscillations is measured
to be 13.41 s. The experiment is repeated with M values of 70.0,
60.0, 50.0, 40.0, and 20.0 g, with corresponding time intervals
for ten oscillations of 12.52, 11.67, 10.67, 9.62, and 7.03 s. (d)
Compute the experimental value for T from each of these
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 37


measurements. (e) Plot a graph of T2 versus M and (f)
determine a value for k from the slope of the linear least-
squares fit through the data points. (g) Compare this value of k
with that obtained in part (b). (h) Obtain a value for ms from
your graph and compare it with the given value of 7.40 g.




65. Review. A light balloon filled with helium of density 0.179
kg/m3 is tied to a light string of length L = 3.00 m. The string is
tied to the ground forming an ―inverted‖ simple pendulum
(Fig. 15.65a). If the balloon is displaced slightly from
equilibrium as in Figure P15.65b and released, (a) show that the
motion is simple harmonic and (b) determine the period of the
motion. Take the density of air to be 1.20 kg/m3. Hint: Use an
analogy with the simple pendulum and see Chapter 14.
Assume the air applies a buoyant force on the balloon but does
not otherwise affect its motion.
                              Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 38




66. Consider the damped oscillator illustrated in Figure 15.20.
The mass of the object is 375 g, the spring constant is 100 N/m,
and b = 0.100 N · s/m. (a) Over what time interval does the
amplitude drop to half its initial value? (b) What If? Over what
time interval does the mechanical energy drop to half its initial
value? (c) Show that, in general, the fractional rate at which the
amplitude decreases in a damped harmonic oscillator is one-
half the fractional rate at which the mechanical energy
decreases.


67.    A block of mass m is connected to two springs of force
constants k1 and k2 in two ways as shown in Figure P15.67. In
both cases, the block moves on a frictionless table after it is
displaced from equilibrium and released. Show that in the two
cases the block exhibits simple harmonic motion with periods

                          m  k1  k2                    m
             (a) T  2                 and (b) T  2
                              k1k2                     k1  k2
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 39




68.    Your thumb squeaks on a plate you have just washed.
Your sneakers squeak on the gym floor. Car tires squeal when
you start or stop abruptly. You can make a goblet sing by
wiping your moistened finger around its rim. When chalk
squeaks on a blackboard, you can see that it makes a row of
regularly spaced dashes. As these examples suggest, vibration
commonly results when friction acts on a moving elastic object.
The oscillation is not simple harmonic motion, but is called
stick-and-slip. This problem models stick-and-slip motion.
      A block of mass m is attached to a fixed support by a
horizontal spring with force constant k and negligible mass
(Fig. P15.68). Hooke’s law describes the spring both in
extension and in compression. The block sits on a long
horizontal board, with which it has coefficient of static friction
s and a smaller coefficient of kinetic friction k. The board
moves to the right at constant speed v. Assume the block
spends most of its time sticking to the board and moving to the
                           Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 40


right with it, so the speed v is small in comparison to the
average speed the block has as it slips back toward the left. (a)
Show that the maximum extension of the spring from its
unstressed position is very nearly given by smg/k. (b) Show
that the block oscillates around an equilibrium position at
which the spring is stretched by kmg/k. (c) Graph the block’s
position versus time. (d) Show that the amplitude of the block’s
motion is

                          A
                                 s  k  mg
                                     k
(e) Show that the period of the block’s motion is
                          2   s   k  mg    m
                     T                      
                                  vk            k
It is the excess of static over kinetic friction that is important for
the vibration. ―The squeaky wheel gets the grease‖ because
even a viscous fluid cannot exert a force of static friction.




69.     Review. A lobsterman’s buoy is a solid wooden
cylinder of radius r and mass M. It is weighted at one end so
that it floats upright in calm seawater, having density . A
passing shark tugs on the slack rope mooring the buoy to a
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 41


lobster trap, pulling the buoy down a distance x from its
equilibrium position and releasing it. (a) Show that the buoy
will execute simple harmonic motion if the resistive effects of
the water are ignored. (b) Determine the period of the
oscillations.


70. Why is the following situation impossible? Your job involves
building very small damped oscillators. One of your designs
involves a spring–object oscillator with a spring of force
constant k = 10.0 N/m and an object of mass m = 1.00 g. Your
design objective is that the oscillator undergo many
oscillations as its amplitude falls to 25.0% of its initial value in
a certain time interval. Measurements on your latest design
show that the amplitude falls to the 25.0% value in 23.1 ms.
This time interval is too long for what is needed in your
project. To shorten the time interval, you double the damping
constant b for the oscillator. This doubling allows you to reach
your design objective.


71. Two identical steel balls, each of mass 67.4 g, are moving in
opposite directions at 5.00 m/s. They collide head-on and
bounce apart elastically. By squeezing one of the balls in a vise
while precise measurements are made of the resulting amount
of compression, you find that Hooke’s law is a good model of
the ball’s elastic behavior. A force of 16.0 kN exerted by each
                           Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 42


jaw of the vise reduces the diameter by 0.200 mm. Model the
motion of each ball, while the balls are in contact, as one-half of
a cycle of simple harmonic motion. Compute the time interval
for which the balls are in contact. (If you solved Problem 57 in
Chapter 7, compare your results from this problem with your
results from that one.)


Challenge Problems


72.     A smaller disk of radius r and mass m is attached
rigidly to the face of a second larger disk of radius R and mass
M as shown in Figure P15.72. The center of the small disk is
located at the edge of the large disk. The large disk is mounted
at its center on a frictionless axle. The assembly is rotated
through a small angle  from its equilibrium position and
released. (a) Show that the speed of the center of the small disk
as it passes through the equilibrium position is
                                                      1/2
                              Rg 1  cos      
                   v  2                         
                          M / m  r / R   2 
                                             2
                                                 
(b) Show that the period of the motion is

                            M  2m  R 2  mr 2 
                                                  1/2

                   T  2                        
                                  2mgR           
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 43




73. An object of mass m1 = 9.00 kg is in equilibrium when
connected to a light spring of constant k = 100 N/m that is
fastened to a wall as shown in Figure P15.73a. A second object,
m2 = 7.00 kg, is slowly pushed up against m1, compressing the
spring by the amount A = 0.200 m (see Fig. P15.73b). The
system is then released, and both objects start moving to the
right on the frictionless surface. (a) When m1 reaches the
equilibrium point, m2 loses contact with m1 (see Fig. P15.73c)
and moves to the right with speed v. Determine the value of v.
(b) How far apart are the objects when the spring is fully
stretched for the first time (the distance D in Fig. P15.48d)?
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 44




74.    Review. Why is the following situation impossible? You are
in the high-speed package delivery business. Your competitor
in the next building gains the right-of-way to build an
evacuated tunnel just above the ground all the way around the
Earth. By firing packages into this tunnel at just the right speed,
your competitor is able to send the packages into orbit around
the Earth in this tunnel so that they arrive on the exact opposite
side of the Earth in a very short time interval. You come up
with a competing idea. Figuring that the distance through the
Earth is shorter than the distance around the Earth, you obtain
permits to build an evacuated tunnel through the center of the
Earth. By simply dropping packages into this tunnel, they fall
downward and arrive at the other end of your tunnel, which is
                           Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 45


in a building right next to the other end of your competitor’s
tunnel. Because your packages arrive on the other side of the
Earth in a shorter time interval, you win the competition and
your business flourishes. Note: An object at a distance r from
the center of the Earth is pulled toward the center of the Earth
only by the mass within the sphere of radius r (the reddish
region in Fig. P15.74). Assume the Earth has uniform density.




75.    A block of mass M is connected to a spring of mass m
and oscillates in simple harmonic motion on a frictionless,
horizontal track (Fig. P15.75). The force constant of the spring is
k, and the equilibrium length is . Assume all portions of the
spring oscillate in phase and the velocity of a segment of the
spring of length dx is proportional to the distance x from the
fixed end; that is, vx = (x/)v. Also, notice that the mass of a
segment of the spring is dm = (m/) dx. Find (a) the kinetic
energy of the system when the block has a speed v and (b) the
period of oscillation.
                           Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 46




76. Review. A system consists of a vertical spring with force
constant k = 1 250 N/m, length L = 1.50 m, and an object of
mass m = 5.00 kg attached to the end (Fig. P15.76). The object is
placed at the level of the point of attachment with the spring
unstretched, at position yi = L, and then it is released so that it
swings like a pendulum. (a) Find the y position of the object at
the lowest point. (b) Will the pendulum’s period be greater or
less than the period of a simple pendulum with the same mass
m and length L? Explain.




77.    A light, cubical container of volume a3 is initially filled
with a liquid of mass density  as shown in Figure P15.77a. The
cube is initially supported by a light string to form a simple
pendulum of length Li, measured from the center of mass of the
                          Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 47


filled container, where Li >> a. The liquid is allowed to flow
from the bottom of the container at a constant rate (dM/dt). At
any time t, the level of the liquid in the container is h and the
length of the pendulum is L (measured relative to the
instantaneous center of mass) as shown in Figure P15.77b. (a)
Find the period of the pendulum as a function of time. (b) What
is the period of the pendulum after the liquid completely runs
out of the container?

				
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